2.1 Insert the function \( f(x)=2 \cos ^{2} x-1 \) \[ \text { Start: }-180^{\circ} \text {; End: } 180^{\circ} \] 0 Step \( 45^{\circ} \)

The function \( f(x)=2 \cos^{2} x - 1 \) is a classic trigonometric transformation that showcases the relationship between cosine and the double-angle identity for sine. This function can also be recognized as \( f(x) = \cos(2x) \). Thus, by plugging in various degrees between -180° and 180°, you can observe the periodic oscillations of the cosine function, revealing its symmetry and repetitive behavior. Each successively calculated value will trace out this wave-like pattern, effectively hitting the peaks and troughs at intervals of 90°. When analyzing or calculating values for \( f(x) \) at each \( 45° \) increment starting from -180° through to 180°, keep an eye out for common pitfalls like miscalculating the cosine values at these angles. Remember that cosine has specific values at key angles: for example, \( \cos(0°) = 1 \), \( \cos(90°) = 0 \), and \( \cos(180°) = -1 \). These calculated outcomes will directly impact your function, turning potential mistakes into learning moments!